No abstract

Главный редактор доктор ф.-м.н. А.К. Арынгазин (Казахстан) Зам. главного редактора А.Т. Акилбеков, доктор ф.-м.н. профессор (Казахстан) Редакционная коллегия Алдонгаров А.А. PhD (Казахстан) Балапанов М.Х. ф.-м.н., проф. (Россия) Бахтизин Р.З. ф.-м.н., проф. (Россия) Гиниятова Ш.Г. кандидат ф.-м.н. (Казахстан) Даулетбекова А.К. кандидат ф.-м.н., PhD (Казахстан) Ержанов К.К. кандидат ф.-м.н., PhD (Казахстан) Жумадилов К.Ш. доктор PhD (Казахстан) Здоровец М. к.ф-м.н.(Казахстан) Кадыржанов К.К. ф.-м.н., проф. (Казахстан) Кайнарбай А.Ж. кандидат ф.-м.н. (Казахстан) Кутербеков К.А. доктор ф.-м.н., проф. (Казахстан) Лущик А.Ч. ф.-м.н., проф. (Эстония) Морзабаев А.К. кандидат ф.-м.н. (Казахстан) Мырзакулов Р.К. доктор ф.-м.н., проф. (Казахстан) Нурахметов Т.Н. доктор ф.-м.н., проф. (Казахстан) Сауытбеков С.С. доктор ф.-м.н., проф. (Казахстан) Тлеукенов С.К. доктор ф.-м.н., проф.

This paper describes the soliton surfaces approach to the Witten-Dijkgraaf-E.Verlinde-H. Verlinde (WDVV) equation. We constructed the surface associated with the WDVV equations using Sym-Tafel formula, which gives a connection between the classical geometry of manifolds immersed in R m and the theory of solitons. The so-called Sym-Tafel formula simplifies the explicit reconstruction of the surface from the knowledge of its fundamental forms, unifies various integrable nonlinearities and enables one to apply powerful methods of the soliton theory to geometrical problems. The soliton surfaces approach is very useful in construction of the so-called integrable geometries. Indeed, any class of soliton surfaces is integrable. Geometrical objects associated with soliton surfaces (tangent vectors, normal vectors, foliations by curves etc.) usually can be identified with solutions to some nonlinear models (spins, chiral models, strings, vortices etc.) [1], [2]. We consider the geometry of surfaces immersed in Euclidean spaces. Such soliton surfaces for the WDVV equation for n = 3 case with an antidiagonal metric η11 = 0 are considered, and first and second fundamental forms of soliton surfaces are found for this case. Also, we study an area of surfaces for the WDVV equation for n = 3 case with an antidiagonal metric η11 = 0.

All external electromagnetic fields are found in which the Klein-Gordon-Fock equation for a charged test particle admits first-order symmetry operators provided that the groups G 3, r £ 3, of motions act transitively on the two-dimensional subspace V 2.

No abstract